11 THE SPACES L1 AND L2 169 (c) Extend the results of Section 13.11 to the case where p is any number such that 1 < p < + oo. (d) Suppose that p>\. Let /e &&,g e && be two functions 2> 0, Show that we have Ni(fg) = Np(/)Nq(#) if and only if there exist two constants a > 0, /3 > 0 such that (x.(f(x))p = f$(g(x))q almost everywhere. (e) Suppose that the measure (JL is bounded and that ft(X) = 1 . For each r > 0 and each measurable function/^ 0 such that/1" is integrable, show that the mapping />h-»Np(/) is an increasing function on the interval ]0, r] (use Holder's inequality). As/>->0, its limit is expf f log \f\ dp\t or 0 if f log \f\ d^= — oo. If the limit is & 0, it follows that /(*) ^ 0 almost everywhere. If exp( f * log |/| dp\ = f |/| dp,, the function |/| is constant almost everywhere. (f) Let /^ 0 be a /^-measurable function, and for each a > 0 let Aa be the set of all x e X such that/(*) ;> a. If/ e JSfg, then apjLt(Aa) g J/p^. Conversely, if the measure /A is bounded and if there exist constants C > 0, e > 0 such that /x(A«) ^ C • a~p~e for all a> 0, then/e &£ (cf. Section 13.9, Problem 3). 13. Let /be a real valued function ^0 defined on R* = ]0, +oo[, which is Lebesgue- measurable and such that |/|p is Lebesgue-integrable (1 < p < -f oo). (a) The function /is integrable on every compact subset of [0, -f oo [, and in particular rx F6c) = /(O dt is defined for all x > 0 (use Holder's inequality). As x tends to 0 Jo or to + oo, the quotient F(x)/x(p~1)/p tends to 0 (same method). (b) Show that the function F(x)/x is /?th-power-integrable on ]0, -f oo [ and that r+00/F(x) Jo -r ("Hardy's inequality"). (Consider first the case where /e Jf(Rf). For each compact /.* interval [a, b] <=• RJ , majorize the integral (F(/)//)p dt, by integrating by parts and Ja using Holder's inequality.) 14. Let p be a real number >0 and let / be a complex ft-measurable function such that (1) |/|p is /x-integrable and (2) etf is ^-integrable for 0</</0. Show that Np(/~1(^r/™ 1)— /)->0 as t-*Q. (Let /== u + w where u and v are real; observe that if we put wt = etu — I — tu, we have wsjs g w,/t for 0 <l ^ ^ t. If 0 < p < 1, use the elementary inequality (# -f 6)p ^ ap -f ^p when <7, b are ^0. Finally, apply (13.8.1) and (13.8.4).) 15. Let (Xn)nzi be a strictly increasing sequence of integers >0, and let h be a rational integer. For each integer N ^ 1 , put Show that the series. For every real number p such that 0 < p < +00, and every mapping /: X ->R, put